Classical measurement of four-vectors In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time, and temperature, and a variational approach to finding a single-component PDF law p ( x ) of a single variable x . However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x , since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario . The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device. The “intrinsic” measurement scenario In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985), or Frieden (2001). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept of the prior scenario to define our unknown information expression. For this purpose we proceed to analyze a particular prior scenario. This is an ideal, 4 N -dimensional measurement scenario – the vector counterpart of the scalar parameter problem of Sec. 1.2.1.
Classical measurement of four-vectors In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time, and temperature, and a variational approach to finding a single-component PDF law p ( x ) of a single variable x . However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x , since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario . The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device. The “intrinsic” measurement scenario In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985), or Frieden (2001). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept of the prior scenario to define our unknown information expression. For this purpose we proceed to analyze a particular prior scenario. This is an ideal, 4 N -dimensional measurement scenario – the vector counterpart of the scalar parameter problem of Sec. 1.2.1.
Classical measurement of four-vectors In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time and temperature, and a variational approach to finding a single-component PDF law p ( x ) of a single variable x. However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario . The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device. The ‘intrinsic’ measurement scenario In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985) or Frieden (1991). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept to define our unknown information expression.
The principle of extreme physical information (EPI) places physics on an information-theoretic footing. It yields many of the fundamental wave equations of physics (of Klein-Gordon, Dirac, etc.). EPI is based upon the measurement of particle four-vectors ${\mathbf{\ensuremath{\theta}}}_{\mathit{n}}$ with random errors xn, n=1,...,N. This leads to an additive Fisher information form whose extremization provides the derivations. However, the model measurement procedure was, in the past, overly restrictive: the real and imaginary parts qn(x), n=1,...,N of the probability amplitude functions for the fluctuations were required to have nonoverlapping support regions. Here we rederive the requisite information form without this restriction. Our model measurement procedure is simply the efficient collection of four-data, that is, N independent measurements of four-coordinates of particles of the field. In effect, each measurement defines a different degree of freedom qn(x) of the scenario. \textcopyright{} 1996 The American Physical Society.
This book defines and develops a unifying principle of physics, that of 'extreme physical information'. The information in question is, perhaps surprisingly, not Shannon or Boltzmann entropy but, rather, Fisher information, a simple concept little known to physicists. Both statistical and physical properties of Fisher information are developed. This information is shown to be a physical measure of disorder, sharing with entropy the property of monotonic change with time. The information concept is applied 'phenomenally' to derive most known physics, from statistical mechanics and thermodynamics to quantum mechanics, the Einstein field equations, and quantum gravity. Many new physical relations and concepts are developed, including new definitions of disorder, time and temperature. The information principle is based upon a new theory of measurement, one which incorporates the observer into the phenomenon that he/she observes. The 'request' for data creates the law that, ultimately, gives rise to the data. The observer creates his or her local reality.
Consider an isolated statistical system specified by a coordinate x and its probability density p(x). A functional of p(x) called ``Fisher information'' can be used to measure the degree of disorder of the system due to the spread in p(x). Fisher information may be minimized, subject to a physical constraint, to attain a temporal equilibrium solution p(x). When the constraint is linear in the mean kinetic energy of the system, the equilibrium solution p(x) often obeys the correct differential equation for the system. In this way, the Schr\"odinger (energy) wave equation, Klein-Gordon equation, Helmholtz wave equation, diffusion equation, Boltzmann law, and Maxwell-Boltzmann law may be derived from one classical principle of disorder. The convergence rate for Fisher information is about that for alternative use of maximum entropy (in problems where both have the same equilibrium solution). This suggests that Fisher information defines an arrow of time. The arrow points in the direction of decreasing accuracy for the determination of the mean, or ideal, value of a parameter.
This book defines and develops a unifying principle of physics, that of 'extreme physical information'. The information in question is, perhaps surprisingly, not Shannon or Boltzmann entropy but, rather, Fisher information, a simple concept little known to physicists. Both statistical and physical properties of Fisher information are developed. This information is shown to be a physical measure of disorder, sharing with entropy the property of monotonic change with time. The information concept is applied 'phenomenally' to derive most known physics, from statistical mechanics and thermodynamics to quantum mechanics, the Einstein field equations, and quantum gravity. Many new physical relations and concepts are developed, including new definitions of disorder, time and temperature. The information principle is based upon a new theory of measurement, one which incorporates the observer into the phenomenon that he/she observes. The 'request' for data creates the law that, ultimately, gives rise to the data. The observer creates his or her local reality.
The principle of minimum Fisher information (MFI) provides a basis for deriving most of the laws of physics.1 The Fisher I is defined as (mean-squares error)-1 in the efficient determination of the classical position θ of a particle. For a one-component pdf p(x) = q2(x), the Fisher information is a one-term functional I = 4 ʃdx[q’(x)]2. Thus, I decreases as q(x) becomes smoother, or as disorder increases, consistent with the second law of thermodynamics. In the above, p(x) was specifically assumed to have a single component q2(x). Then MFI gives a real-only wave equation. Here we consider multicomponent cases p(x) = q12(x) + … + qN2(x), where the qn(x) are independent degrees of freedom. We seek the Fisher I for the case N = 2 of nonrelativistic quantum mechanics or of scalar optics. A benchmark statistical situation arises when the N components of qn(x) do not overlap. Then their informations add, and the resulting mean-squares error value is efficient, i.e. an absolute minimum.